Question: Vanessa is 12 years older than Emily. Vanessa and Emily first met 3 years ago. Fourteen years ago, Vanessa was 4 times older than Emily. How old is Vanessa now?
Answer: We can use the given information to write down two equations that describe the ages of Vanessa and Emily. Let Vanessa's current age be $v$ and Emily's current age be $e$ The information in the first sentence can be expressed in the following equation: $v = e + 12$ Fourteen years ago, Vanessa was $v - 14$ years old, and Emily was $e - 14$ years old. The information in the second sentence can be expressed in the following equation: $v - 14 = 4(e - 14)$ Now we have two independent equations, and we can solve for our two unknowns. Because we are looking for $v$ , it might be easiest to solve our first equation for $e$ and substitute it into our second equation. Solving our first equation for $e$ , we get: $e = v - 12$ . Substituting this into our second equation, we get the equation: $v - 14 = 4($ $(v - 12)$ $ -$ $ 14)$ which combines the information about $v$ from both of our original equations. Simplifying the right side of this equation, we get: $v - 14 = 4v - 104$ Solving for $v$ , we get: $3 v = 90$ $v = 30$.